Lévy's continuity theorem

In probability theory, Lévy’s continuity theorem, or Lévy's convergence theorem,{{cite book

| last = Williams | first = D. | authorlink = David Williams (mathematician)

| title = Probability with Martingales

| year = 1991

| publisher = Cambridge University Press

| isbn = 0-521-40605-6 |at=section 18.1

}} named after the French mathematician Paul Lévy, connects convergence in distribution of the sequence of random variables with pointwise convergence of their characteristic functions.

This theorem is the basis for one approach to prove the central limit theorem and is one of the major theorems concerning characteristic functions.

Statement

Suppose we have

{{unordered list

|1= a sequence of random variables $\{X_n\}_{n=1}^\infty$ , not necessarily sharing a common probability space,

|2= the sequence of corresponding characteristic functions $\{\varphi_n\}_{n=1}^\infty$ , which by definition are

: $\varphi_n(t) = \operatorname{E} \left[ e^{itX_n} \right] \quad \forall t\in\mathbb{R},\ \forall n\in\mathbb{N},$

where $\operatorname{E}$ is the expected value operator.

}}

If the sequence of characteristic functions converges pointwise to some function $\varphi$

: $\varphi_n(t)\to\varphi(t) \quad \forall t\in\mathbb{R},$

then the following statements become equivalent:

{{unordered list

|1= $X_n$ converges in distribution to some random variable X

: $X_n\ \xrightarrow{\mathcal D}\ X,$

i.e. the cumulative distribution functions corresponding to random variables converge at every continuity point of the c.d.f. of X;

|2= $\{X_n\}_{n=1}^\infty$ is tight:

: $\lim_{x\to\infty}\left( \sup_n \operatorname{P}\big[\, |X_n|>x \,\big]\right) = 0;$

|3= $\varphi(t)$ is a characteristic function of some random variable X;

|4= $\varphi(t)$ is a continuous function of t;

|5= $\varphi(t)$ is continuous at t = 0.

}}

Proof

Rigorous proofs of this theorem are available.{{cite book |last=Fristedt |first=B. E. |last2=Gray |first2=L. F. |year=1996 |title=A modern approach to probability theory |publisher=Birkhäuser |location=Boston |isbn=0-8176-3807-5 |at=Theorems 14.15 and 18.21 }}

References

Category:Theorems in probability theory

Category:Theorems in statistics

Category:Paul Lévy (mathematician)